Correct Answer - Option 4 :
\(\frac{25}{24}\)
Concept:
\(\rm 2\sin^{-1}x= \sin^{-1}[{2x}{\sqrt{1-x^{2}}}]\)
\(\rm \sin^{-1}x= \ cosec^{-1}\frac{1}{x}\)
Calculation:
We know that, \(\rm 2\sin^{-1}x= \sin^{-1}[{2x}{\sqrt{1-x^{2}}}]\)
∴ \(\rm \left \{ 2\sin^{-1} \left ( \frac{3}{5} \right )\right \}\) = \(\rm \sin^{-1}\left [ 2\times \frac{3}{5}\times\sqrt{1-\frac{9}{25}} \right ]\)
⇒ \(\rm \left \{ 2\sin^{-1}\left ( \frac{3}{5} \right )\right \}\) = \(\rm \sin^{-1} \left ( \frac{24}{25} \right )\)
As we know that, \(\rm \sin^{-1}x= \ cosec^{-1}\frac{1}{x}\)
= \(\rm \ cosec\left \{ \ cosec^{-1}\left ( \frac{25}{24} \right )\right \}\)
= \(\frac{25}{24}\) .
The correct option is 4.
